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Finite reflection groups
q, t-Fuß-Catalan numbers for real reflection groups
Algebraic Combinatorics – the extended Shi arrangement
Combinatorial Algebra – rational Cherednik algebras
q, t-Fuß-Catalan numbers for complex reflection groups
Finite real reflection groups
Let V be a finite-dimensional real vector space.
I A (finite) real reflection group
W = 〈t1, . . . , t`〉 ⊆ O(V )
is a finite group generated by reflections.
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I The following list of root systems determine (up toisomorphisms) the irreducible finite real reflection groups:
An−1 (symmetric group),
Bn (group of signed permutations),
Dn (group of even-signed permutations),
I2(k) (dihedral group of order 2k) and
H3,H4,F4,E6,E7,E8 (exceptional groups).
The most classical example of a reflection group is the symmetricgroup Sn of all permutations of n letters.
123 = (), 132 = (23), 213 = (12),
231 = (123), 312 = (132), 321 = (13).
This group can be seen as the reflection group of type An−1:
transposition ←̃→ reflection
(i , j) = ei ↔ ej
simple transposition ←̃→ simple reflection
(i , i + 1) = ei ↔ ei+1.
Finite complex reflection groups
Let V be a finite-dimensional complex vector space.
I A complex reflection s ∈ U(V )
i. has finite order andii. its fixed-point space has codimension 1.
I A (finite) complex reflection group
W = 〈t1, . . . , t`〉 ⊆ O(V )
is a finite group generated by complex reflections.
Irreducible complex reflection groups are determined by thefollowing types:
G (m, p, n) with p|m of order mnn!/p,
G4 − G37 34 exceptional types.
(Shephard–Todd, Chevalley, 1950’s)
t t
tt
tt
1 = ζ5
ζ
ζ2
ζ3
ζ4
I+
W
-
�
I It acts on C by multiplication of a primitive root of unity ζ.
I In real reflection groups, any reflection t has order two andthere is a 1 : 1-correspondence{
reflections}←̃→
{reflecting hyperplanes
}t ↔ Hα.
I In complex reflection groups, any reflection t has orderk ≥ 2 and there is a correspondence{
reflections}←̃→
{reflecting hyperplanes
}t, t2, . . . , tk−1 ↔ Hα.
For a permutation σ ∈ Sn, define a diagonal action onC[x, y] := C[x1, y1, . . . , xn, yn] by σ(xi ) := xσ(i), σ(yi ) := yσ(i).E.g.,
231(2x1x2y22 y3) = 2x2x3y2
3 y1.
A polynomial f ∈ C[x, y] is called
I invariant if σ(f ) = f ,
I alternating if σ(f ) = sgn(σ) f .
Example
x1y2 + x2y1 is invariant, x1y2 − x2y1 is alternating.
−Generalization
Let W be a real reflection group acting on V . Thecontragredient action of W on V ∗ = Hom(V ,C) is given by
ω(ρ) := ρ ◦ ω−1.
This gives an action of W on the symmetric algebra S(V ∗) = C[x]and ’doubling up’ this action gives a diagonal action onC[x, y] := C[V ⊕ V ].
A polynomial f ∈ C[x, y] is called
I invariant if ω(f ) = f ,
I alternating if ω(f ) = det(ω) f .
q, t-Fuß-Catalan numbers as a bigraded Hilbert series
Let W be a reflection group now acting on C[x, y] and let
A := 〈 alternating polynomials 〉 E C[x, y].
Define the W -module M(m)(W ) to be minimal generating spaceof the ideal Am,
M(m)(W ) := Am/〈x, y〉Am ∼= CB,
where B is any homogeneous minimal generating set for Am.
M(m)(W ) sits inside a larger W -module DR(m)(W ) as its isotypiccomponent,
M(m)(W ) ∼= edet
(DR(m)(W )
).
q, t-Fuß-Catalan numbers as a bigraded Hilbert series
DefinitionFor any real reflection group W , define q, t-Fuß-Catalannumbers to be the bigraded Hilbert series of M(m)(W ),
Cat(m)(W ; q, t) := H(M(m)(W ); q, t)
=∑f ∈B
qdegx(f )tdegy(f ).
I Cat(m)(W ; q, t) is a symmetric polynomial in q and t,
I it reduces in type An−1 to the classical q, t-Fuß-Catalannumbers,
Cat(m)(Sn; q, t) = Cat(m)n (q, t)
introduced by Haiman in the 1990’s.
Example: Cat(1)(S3; q, t)
For W = S3, one can show that
M(1)(S3) = C{
∆{(1,0),(2,0)},∆{(1,0),(1,1)},∆{(0,1),(1,1)},
∆{(0,1),(0,2)},∆{(1,0),(0,1)}
},
where
∆{(i1,j1),(i2,j2)}(x, y) := det
1 1 1
x i11 y j1
1 x i12 y j1
2 x i13 y j1
3
x i21 y j2
1 x i22 y j2
2 x i23 y j2
3
is the generalized Vandermonde determinant. This gives
Cat(1)(S3; q, t) = H(M(1)(S3); q, t)
= q3 + q2t + qt2 + t3 + qt.
A conjectured formula for the dimension of M (m)(W )
Computations of the dimensions of M(m)(W ) were the firstmotivation for further investigations:
Conjecture
Let W be a real reflection group. Then
Cat(m)(W ; 1, 1) =∏̀i=1
di + mh
di,
where
I ` is the rank of W ,
I h is the Coxeter number and
I d1, . . . , d` are its degrees.
I These numbers, called Fuß-Catalan numbers, count severalcombinatorial objects, e.g.,
I positive regions in the generalized Shi arrangement(Athanasiadis, Postnikov),
I m-divisible non-crossing partitions (Armstrong, Bessis,Reiner),
I facets in the generalized Cluster complex (Fomin, Reading,Zelevinsky).
I They reduce for m = 1 to the well-known Catalan numbersassociated to real reflection groups:
An−1 Bn Dn1
n+1
(2nn
) (2nn
) (2nn
)−(2(n−1)
n−1
)I2(k) H3 H4 F4 E6 E7 E8
k + 2 32 280 105 833 4160 25080
The classical q, t-Fuß-Catalan numbers
In type A, Cat(m)(Sn; q, t) occurred within the past 15 years invarious fields of mathematics:
I Hilbert series of space of diagonal coinvariants (Haiman),
I complicated rational function in the context of modifiedMacdonald polynomials (Garsia, Haiman),
I Hilbert series of some cohomology module in the theory ofHilbert schemes of points in the plane (Haiman),
I they have a conjectured combinatorial interpretation in termsof two statistics on partitions fitting inside the partitionµ := ((n − 1)m, . . . , 2m,m),
Cat(m)n (q, t) =
∑λ⊆µ
qarea(λ)tbounce(λ).
I Proved for m = 1 (Garsia, Haglund) and for t = 1 (Haiman).
The extended Shi arrangement
Let W be a crystallographic reflection group.
Shi(m)(W ) is defined to be the collection of (translates of thereflecting) hyperplanes in V given by{
Hkα : α ∈ Φ+,−m < k ≤ m
},
where Hkα = {x ∈ V : (x , α) = k}.
I A region of Shi(m)(W ) is a connected component of itscomplement.
Remark
Coxeter arrangement ⊆ extended Shi arrangement.
The extended Shi arrangement
Let W be a crystallographic reflection group.
Theorem (Yoshinaga)
The number of regions in Shi(m)(W ) is given by
(mh + 1)`.
Theorem (Athanasiadis)
The number of positive regions in Shi(m)(W ) – regions which liein the fundamental chamber of the associated Coxeterarrangement – is given by
∏̀i=1
di + mh
di.
Example: Shi(1)(A2)
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H1α2
H0α3
H1α3
H0α1
H1α1∣∣∣{ regions
}∣∣∣ = 16 = (1 · 3 + 1)2,∣∣∣{ positive regions}∣∣∣ = 5 =
5 · 62 · 3
.
Example: Shi(1)(A2) and Cat(1)(A2; q)
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Cat(1)(A2; q) =∑
qcoh(R) = 1 + 2q + q2 + q3.
Specialization t = 1.
Conjecture
Let W be a crystallographic reflection group. Then
Cat(m)(W ; q, 1) =∑
qcoh(R),
where the sum ranges over all regions of Shi(m)(Φ) which lie in thefundamental chamber of the associated Coxeter arrangementand where coh denotes the coheight statistic.
I The conjecture is known to be true for type A,
I was validated by computations for several types.
Specialization t = q−1.
Conjecture
Let W be a reflection group. Then
Cat(m)(W ; q, q−1) = q−mN∏̀i=1
[di + mh]q[di ]q
,
where
I N is the number of reflections in W ,
I [n]q := qn−1 + . . .+ q + 1 is the usual q-analogue of n.
TheoremLet W be the dihedral group of type I2(k). Then
Cat(m)(W ; q, t) =m∑
j=0
qm−j tm−j [ jk + 1]q,t ,
where
[n]q,t := qn−1 + qn−2t + . . .+ qtn−2 + tn−1.
TheoremAll shown conjectures hold for the dihedral groups.
Let W be a real reflection group. The rational Cherednik algebra
Hc = Hc(W )
is an associative algebra generated by
V ,V ∗,W ,
subject to defining relations depending on a rational parameterc , such that
I the polynomial rings C[V ],C[V ∗] and
I the group algebra CW
are subalgebras of Hc .
A simple Hc-module
For c = m + 1h there exists a unique simple Hc -module L which
carries a natural filtration.
Theorem (Berest, Etingof, Ginzburg)
Let W be a real reflection group. Then
H(gr(L); q) = q−mN [mh + 1]`q,
H(e(gr(L)); q) = q−mN∏̀i=1
[di + mh]q[di ]q
,
where gr(L) is the associated graded module of L and wheree(gr(L)) is its trivial component.
The connection between L and the space of generalizeddiagonal coinvariants
TheoremLet W be a real reflection group and let
DR(m)(W ) = Am−1/IAm−1
be the generalized diagonal coinvariants graded by degree in xminus degree in y. Then there exists a natural surjection ofgraded modules,
DR(m)(W )⊗ det� gr(L),
where det denotes the determinantal representation.
RemarkThis theorem generalizes a theorem by Gordon, who proved them = 1 case, following mainly his approach.
The connection between L and the space of generalizeddiagonal coinvariants
Conjecture (Haiman)
The W -stable kernel of the surjection in the previous theorem doesnot contain a copy of the trivial representation.
Corollary
If the previous conjecture holds, then
M(m)(W ) ∼= e(gr(L))
as graded modules. In particular,
Cat(W ; q, q−1) = q−mN∏̀i=1
[di + mh]q[di ]q
.
The cyclic group Ck = 〈ζ〉 would act on C[x , y ] := C[C⊕ C] by
ζ(xayb) = ζa · xa ζb · yb = ζa+b · xayb.
This would give
C[x , y ]Ck = span{
xayb : a + b ≡ 0 mod k},
C[x , y ]det = span{
xayb : a + b ≡ 1 mod k}
= xC[x , y ]Ck + yC[x , y ]Ck ,
C[x , y ]det−1
= span{
xayb : a + b ≡ k − 1 mod k}
=∑
i+j=k−1
x iy jC[x , y ]Ck .
C[x , y ]det = xC[x , y ]Ck + yC[x , y ]Ck ,
C[x , y ]det−1
=∑
i+j=k−1
x iy jC[x , y ]Ck .
We would have two possible choices to define q, t-Catalan numbersfor the cyclic group Ck :
Cat(1)(Ck ; q, t) = q + t or
Cat(1)(Ck ; q, t) = qk−1 + qk−2t + . . .+ qtk−2 + tk−1.
I Both choices would be in contradiction to the previouslyshown conjectures!
−Generalization
Let W be a real reflection group acting on V .The contragredient action of W on V ∗ = Hom(V ,C) is given by
ω(ρ) := ρ ◦ ω−1.
This gives an action of W on the symmetric algebra S(V ∗) = C[x]and ‘doubling up’ this action gives a diagonal action on
C[x, y] := C[V ⊕ V ].
A polynomial f ∈ C[x, y] is called
I invariant if ω(f ) = f for all ω ∈W ,
I alternating if ω(f ) = det(ω) f for all ω ∈W .
−Generalization
Let W be a complex reflection group acting on V .The contragredient action of W on V ∗ = Hom(V ,C) is given by
ω(ρ) := ρ ◦ ω−1.
This gives an action of W on the symmetric algebra S(V ∗) = C[x]and ’doubling up’ this action gives a diagonal action on
C[x, y] := C[V ⊕ V ∗].
A polynomial f ∈ C[x, y] is called
I invariant if ω(f ) = f for all ω ∈W ,
I alternating if ω(f ) = det(ω) f for all ω ∈W .
The cyclic group Ck = 〈ζ〉 would act on C[x , y ] := C[C⊕ C] by
ζ(xayb) = ζa · xa ζb · yb = ζa+b · xayb.
The cyclic group Ck = 〈ζ〉 acts on C[x , y ] := C[C⊕ C∗] by
ζ(xayb) = ζa · xa ζ−b · yb = ζa−b · xayb.
The cyclic group Ck = 〈ζ〉 acts on C[x , y ] := C[C⊕ C∗] by
ζ(xayb) = ζa · xa ζ−b · yb = ζa−b · xayb.This gives
C[x , y ]Ck = span{
xayb : a ≡ b mod k},
C[x , y ]det = span{
xayb : a + 1 ≡ b mod k}
= xC[x , y ]Ck + yk−1C[x , y ]Ck ,
C[x , y ]det−1
= span{
xayb : a ≡ b + 1 mod k}
= xk−1C[x , y ]Ck + yC[x , y ]Ck .
The cyclic group Ck = 〈ζ〉 acts on C[x , y ] := C[C⊕ C∗] by
ζ(xayb) = ζa · xa ζ−b · yb = ζa−b · xayb.
C[x , y ]det = xC[x , y ]Ck + yk−1C[x , y ]Ck ,
C[x , y ]det−1
= xk−1C[x , y ]Ck + yC[x , y ]Ck .
Now, we have (beside interchanging the roles of q and t) only thefollowing choice:
Cat(1)(Ck ; q, t) := q + tk−1.
The cyclic group Ck = 〈ζ〉 acts on C[x , y ] := C[C⊕ C∗] by
ζ(xayb) = ζa · xa ζ−b · yb = ζa−b · xayb.
Now, we have (beside interchanging the roles of x and y) only thefollowing choice:
Cat(1)(Ck ; q, t) := q + tk−1.
I The q-power equals the number of reflecting hyperplanesN∗ = 1 and the t-power equal the number of reflectionsN = k − 1. We have
qN Cat(1)(Ck ; q, q−1) = qN∗Cat(1)(Ck ; q−1, q)
= 1 + qk = [2k]q/[k]q.
Conjecture
Let W be a well-generated complex reflection group. Then
qmN Cat(m)(W ; q, q−1) = qmN∗Cat(m)(W ; q−1, q)
=∏̀i=1
[di + mh]q[di ]q
,
where
I N is the number of reflections in W and where
I N∗ is the number of reflecting hyperplanes.